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First-order linear equations

Problem 6.125 · medium Video

Solve \( \displaystyle y' + 1y = 3 e^{- x} \) with \( \displaystyle y(0) = -1 \).

The equation is linear in standard form; the integrating factor is e^(∫1 dx) = e^(1x).
\[ \frac{d}{d x} Y{\left(x \right)} e^{x} = Y{\left(x \right)} e^{x} + e^{x} \frac{d}{d x} Y{\left(x \right)} \]
Multiplying by e^(ax) turns the left side into (e^(ax) y)'.✓ Proved
\[ \int 3\, dx = 3 x \]
Integrate the right side.✓ Proved
Setting x = 0 and y = -1 fixes the constant of integration: C = -1.
\[ \left(3 x - 1\right) e^{- x} + \frac{d}{d x} \left(3 x - 1\right) e^{- x} = 3 e^{- x} \]
The solution satisfies the equation.✓ Proved
\[ -1 \]
And the initial condition.

Answer \( y = \left(3 x - 1\right) e^{- x} \)

The slope field of y' = -y + 3e^(-x): short segments showing the slope at each point. Several solution curves follow the field; the one through (0, -1) is highlighted, y = (3x - 1)·e^(-x).

Lines: 3 proved, 3 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	Not checked	—	a sentence; read, not computed
5	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
6	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	sympy.checkodesol substitutes the solution back; y(0) matches

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution skips the crucial step of integrating the right-hand side (3*e^x) to find the general solution before applying the initial condition. It jumps from the setup to the final answer, making the determination of C = -1 unjustified and opaque.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution skips the crucial step of integrating the right-hand side (3*e^x) to find the general solution before applying the initial condition. It jumps from the setup to the final answer, making the determination of C = -1 unjustified and opaque.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution skips the crucial step of integrating the right-hand side (which is 3*e^x, not 3) and determining the constant of integration. It jumps from the derivative form to the final answer without showing the integration of 3*e^x to get 3x*e^x + C, nor does it correctly derive C=-1 from the intermediate step. The sentence in step 4 claims C=-1 is fixed by the initial condition, but the algebraic steps connecting the integrated form to the final answer are missing, making the derivation incomplete and logically disjointed.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The integrating factor is calculated as e^x, but the subsequent steps (integration and final solution) incorrectly use e^{-x} as the integrating factor or fail to divide by it properly. Specifically, step 3 integrates 3 instead of 3e^x, and step 4 derives C=-1 based on the incorrect form y = 3x + C rather than the correct y = (3x + C)e^{-x}. The stated answer is actually correct for the problem, but the derivation shown is mathematically inconsistent and wrong.
gpt-oss:20b: pass 2026-09-27

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/linear_first_order, checked 2026-10-13 with SymPy 1.14.0.